Crusells-Girona 2017a - Revision history

Scipediacontent at 11:42, 28 April 2017

2017-04-28T11:42:51Z

Scipediacontent at 11:42, 28 April 2017

2017-04-28T11:42:33Z

Mcrusells at 23:48, 27 April 2017

2017-04-27T23:48:08Z

Mcrusells at 23:34, 27 April 2017

2017-04-27T23:34:38Z

Mcrusells: Mcrusells moved page Draft Crusells-Girona 842065898 to Crusells-Girona 2017a

2017-04-27T23:27:25Z

Mcrusells moved page Draft Crusells-Girona 842065898 to Crusells-Girona 2017a

Mcrusells: Created page with "==NONLINEAR STATIC AND DYNAMIC ANALYSIS OF MIXED CABLE ELEMENTS== '''MIQUEL CRUSELLS-GIRONA $^{}$ , FILIP C. FILIPPOU $^{}$ AND ROBERT L. TAYLOR $^..."$

2017-04-27T23:27:24Z

Created page with "==NONLINEAR STATIC AND DYNAMIC ANALYSIS OF MIXED CABLE ELEMENTS== '''MIQUEL CRUSELLS-GIRONA<math>^{*}</math>, FILIP C. FILIPPOU<math>^{*}</math> AND ROBERT L. TAYLOR<math>^..."

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← Older revision		Revision as of 11:42, 28 April 2017
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	This paper presents a family of finite elements for the nonlinear static and dynamic analysis of cables based on a mixed variational formulation in curvilinear coordinates and finite deformations. This formulation identifies stress measures, in the form of axial forces, and conjugate deformation measures for the nonlinear catenary problem. The continuity requirements lead to two distinct implementations: one with a continuous axial force distribution and one with a discontinuous. Two examples from the literature on nonlinear cable analysis are used to validate the proposed formulation for St Venant-Kirchhoff elastic materials. These studies show that displacements and axial forces are captured with high accuracy for both the static and the dynamic case.		This paper presents a family of finite elements for the nonlinear static and dynamic analysis of cables based on a mixed variational formulation in curvilinear coordinates and finite deformations. This formulation identifies stress measures, in the form of axial forces, and conjugate deformation measures for the nonlinear catenary problem. The continuity requirements lead to two distinct implementations: one with a continuous axial force distribution and one with a discontinuous. Two examples from the literature on nonlinear cable analysis are used to validate the proposed formulation for St Venant-Kirchhoff elastic materials. These studies show that displacements and axial forces are captured with high accuracy for both the static and the dynamic case.
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−	~~'''keywords''' Cables, Mixed FEM, Weak Compatibility, Nonlinear Dynamics~~

	==1 INTRODUCTION==		==1 INTRODUCTION==

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−	~~==NONLINEAR STATIC AND DYNAMIC ANALYSIS OF MIXED CABLE ELEMENTS==~~
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−	~~'''MIQUEL CRUSELLS-GIRONA<math>^{*}</math>,~~
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−	~~FILIP C. FILIPPOU<math>^{}</math> AND ROBERT L. TAYLOR<math>^{}</math>'''~~
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−	{\|
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−	~~\|<math>^{*}</math> Department of Civil and Environmental Engineering~~
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−	~~\| University of California, Berkeley~~
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−	~~\| 94720 Berkeley, CA, USA~~
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−	~~\| e-mail: [mailto:miquel.crusells@berkeley.edu miquel.crusells@berkeley.edu]~~
−	\|}
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	==Abstract==		==Abstract==

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 The first example investigates the stability of a 3d cable supported by a pulley that was previously studied by Impollonia ''et al'' <span id='citeF-6'></span>[[#cite-6|[6]]]. The structural model, whose geometric and material properties are shown in Table&nbsp;[[#table-1|1]], consists of a cable anchored at both ends and supported by an intermediate roller. In this case, the inertia forces in Eq.&nbsp;[[#eq-21|21]] and the mass term of the stiffness are not considered as the problem is analyzed in a static manner.
 {|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;"
 |+ <span id='table-1'></span>Table. 1 <span style="text-align: center; font-size: 75%;">Geometry and Material Properties for Example 1.</span>
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 Following the form finding procedure by Argyris ''et al'' <span id='citeF-2'></span><span id='citeF-7'></span>[[#cite-2|[2,7]]], the analysis starts from a straight reference configuration, and imposes a displacement <math display="inline">\mathbf{u}=(-200,0,50)</math> m at the right support and a pair of displacements <math display="inline">u_2 = 50</math> m and <math display="inline">u_3= 100</math> m at the intermediate roller. Because friction is not considered, the jump in the Cauchy axial force at the roller support must be zero. As a result, the problem is solved by iterating over the curvilinear coordinate <math display="inline">\xi ^1</math> so that the jump in the Cauchy axial force at the pulley becomes zero.
 {|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;"
 |+ <span id='table-2'></span>Table. 2 <span style="text-align: center; font-size: 75%;">Results for Example 1 from different studies.</span>
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 The second example studies the large-amplitude free vibration of two cables with diffe-rent sag/span ratio that were investigated by Srinil ''et al'' <span id='citeF-8'></span>[[#cite-8|[8]]]. The structural model consists of a cable anchored at both ends and spanning  850 m in both cases. Table&nbsp;[[#table-3|3]] summarizes the geometric and material properties of the cables denoted by C1 and C2.
 {|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;"
 |+ <span id='table-3'></span>Table. 3 <span style="text-align: center; font-size: 75%;">Geometry and Material Properties for Example 1.</span>
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 | colspan="2" | '''Figure 3:''' <span style="text-align: center; font-size: 75%;">Normalized vertical eigenvectors for Example 2.</span>
 |}
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-[h!]
 {|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;"
 |+ <span id='table-4'></span>Table. 4 <span style="text-align: center; font-size: 75%;">Results for equilibrium configurations and natural vibration in Example 2.</span>

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-| style="text-align: center;" | <math>\left\{ \begin{array}{rl}\displaystyle G^{11}\frac{d\mathbf{u}}{d\xi ^1} \cdot \left(\mathbf{G}_1+\frac{1}{2}\frac{d\mathbf{u}}{d\xi ^1}\right)-\mathrm{\hat{E}}=0 & \mathrm{in} \; \Omega=(0,L) \\[3.5mm] \displaystyle \frac{d}{dS}\left(\sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1 \right)+\mathbf{\hat{W}}=\rho _0 \mathbf{a}  & \mathrm{in} \; \Omega=(0,L)\\[] [4mm] \mathrm{\hat{N}}-\Psi (\mathrm{\hat{E}})=0 & \mathrm{in} \; \Omega=(0,L) \\[] [1.5mm] \mathbf{u} = \mathbf{\bar{u}} & \mathrm{on} \; \Gamma _u \\[1mm] \displaystyle \sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1= \mathbf{\bar{T}} & \mathrm{on} \; \Gamma _q \end{array} \right. </math>
+| style="text-align: center;" | <math>\left\{ \begin{array}{rl}\displaystyle G^{11}\frac{d\mathbf{u}}{d\xi ^1} \cdot \left(\mathbf{G}_1+\frac{1}{2}\frac{d\mathbf{u}}{d\xi ^1}\right)-\mathrm{\hat{E}}=0 & \mathrm{in} \; \Omega=(0,L) \\[3.5mm] \displaystyle \frac{d}{dS}\left(\sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1 \right)+\mathbf{\hat{W}}=\rho _0 \mathbf{a}  & \mathrm{in} \; \Omega=(0,L)\\[] \mathrm{\hat{N}}-\Psi (\mathrm{\hat{E}})=0 & \mathrm{in} \; \Omega=(0,L) \\[] \mathbf{u} = \mathbf{\bar{u}} & \mathrm{on} \; \Gamma _u \\[1mm] \displaystyle \sqrt{G^{11}}\mathrm{\hat{N}}\mathbf{g}_1= \mathbf{\bar{T}} & \mathrm{on} \; \Gamma _q \end{array} \right. </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)

← Older revision	Revision as of 23:27, 27 April 2017
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Crusells-Girona 2017a - Revision history

Scipediacontent at 11:42, 28 April 2017

Scipediacontent at 11:42, 28 April 2017

Mcrusells at 23:48, 27 April 2017

Mcrusells at 23:34, 27 April 2017

Mcrusells: Mcrusells moved page Draft Crusells-Girona 842065898 to Crusells-Girona 2017a

Mcrusells: Created page with "==NONLINEAR STATIC AND DYNAMIC ANALYSIS OF MIXED CABLE ELEMENTS== '''MIQUEL CRUSELLS-GIRONA^{*}, FILIP C. FILIPPOU^{*} AND ROBERT L. TAYLOR^..."

Mcrusells: Created page with "==NONLINEAR STATIC AND DYNAMIC ANALYSIS OF MIXED CABLE ELEMENTS== '''MIQUEL CRUSELLS-GIRONA $^{}$ , FILIP C. FILIPPOU $^{}$ AND ROBERT L. TAYLOR $^..."$